Remark Let $X$ and $Y$ be topological spaces. Then the join $X \star Y$ of Construction is equipped with a pair of maps $\iota _{X}: X \hookrightarrow X \star Y$ and $\iota _{Y}: Y \hookrightarrow X \star Y$. It is not difficult to see that these maps are closed embeddings: that is, they induce homeomorphisms from $X$ and $Y$ onto closed subsets of $X \star Y$. We will generally abuse notation by identifying $X$ and $Y$ with their images under $\iota _{X}$ and $\iota _{Y}$, respectively.